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Theorem llynlly 22774
Description: A locally 𝐴 space is n-locally 𝐴: the "n-locally" predicate is the weaker notion. (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
llynlly (𝐽 ∈ Locally 𝐴𝐽 ∈ 𝑛-Locally 𝐴)

Proof of Theorem llynlly
Dummy variables 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 llytop 22769 . 2 (𝐽 ∈ Locally 𝐴𝐽 ∈ Top)
2 llyi 22771 . . . . 5 ((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) → ∃𝑢𝐽 (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))
3 simpl1 1191 . . . . . . . 8 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Locally 𝐴)
43, 1syl 17 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝐽 ∈ Top)
5 simprl 769 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢𝐽)
6 simprr2 1222 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑦𝑢)
7 opnneip 22416 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑢𝐽𝑦𝑢) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
84, 5, 6, 7syl3anc 1371 . . . . . 6 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢 ∈ ((nei‘𝐽)‘{𝑦}))
9 simprr1 1221 . . . . . . 7 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢𝑥)
10 velpw 4563 . . . . . . 7 (𝑢 ∈ 𝒫 𝑥𝑢𝑥)
119, 10sylibr 233 . . . . . 6 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢 ∈ 𝒫 𝑥)
128, 11elind 4152 . . . . 5 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → 𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥))
13 simprr3 1223 . . . . 5 (((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) ∧ (𝑢𝐽 ∧ (𝑢𝑥𝑦𝑢 ∧ (𝐽t 𝑢) ∈ 𝐴))) → (𝐽t 𝑢) ∈ 𝐴)
142, 12, 13reximssdv 3167 . . . 4 ((𝐽 ∈ Locally 𝐴𝑥𝐽𝑦𝑥) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
15143expb 1120 . . 3 ((𝐽 ∈ Locally 𝐴 ∧ (𝑥𝐽𝑦𝑥)) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
1615ralrimivva 3195 . 2 (𝐽 ∈ Locally 𝐴 → ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴)
17 isnlly 22766 . 2 (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥𝐽𝑦𝑥𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽t 𝑢) ∈ 𝐴))
181, 16, 17sylanbrc 583 1 (𝐽 ∈ Locally 𝐴𝐽 ∈ 𝑛-Locally 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087  wcel 2106  wral 3062  wrex 3071  cin 3907  wss 3908  𝒫 cpw 4558  {csn 4584  cfv 6493  (class class class)co 7351  t crest 17256  Topctop 22188  neicnei 22394  Locally clly 22761  𝑛-Locally cnlly 22762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-top 22189  df-nei 22395  df-lly 22763  df-nlly 22764
This theorem is referenced by:  llyssnlly  22775  efmndtmd  23398
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